Currently,
R and
S must both be polynomial rings over the same base field.
This function first checks to see whether M will be a finitely generated R-module via F. If not, an error message describing the codimension of M/(vars of S)M is given (this is equal to the dimension of R if and only if M is a finitely generated R-module.
Assuming that it is, the push forward
F_*(M) is computed. This is done by first finding a presentation for
M in terms of a set of elements that generates
M as an
S-module, and then applying the routine
coimage to a map whose target is
M and whose source is a free module over
R.
Example: The Auslander-Buchsbaum formula
Let's illustrate the Auslander-Buchsbaum formula. First construct some rings and make a module of projective dimension 2.
i1 : R4 = ZZ/32003[a..d];
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i2 : R5 = ZZ/32003[a..e];
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i3 : R6 = ZZ/32003[a..f];
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i4 : M = coker genericMatrix(R6,a,2,3)
o4 = cokernel | a c e |
| b d f |
2
o4 : R6-module, quotient of R6
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i5 : pdim M
o5 = 2
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Create ring maps.
i6 : G = map(R6,R5,{a+b+c+d+e+f,b,c,d,e})
o6 = map(R6,R5,{a + b + c + d + e + f, b, c, d, e})
o6 : RingMap R6 <--- R5
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i7 : F = map(R5,R4,random(R5^1, R5^{4:-1}))
o7 = map(R5,R4,{- 6613a + 9106b - 5774c - 12201d - 8915e, 15955a - 6807b + 3795c - 11417d + 15876e, 4799a - 2292b - 4461c + 11786d + 12397e, 12278a - 14019b - 14565c - 10162d + 10007e})
o7 : RingMap R5 <--- R4
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The module M, when thought of as an R5 or R4 module, has the same depth, but since depth M + pdim M = dim ring, the projective dimension will drop to 1, respectively 0, for these two rings.
i8 : P = pushForward(G,M)
o8 = cokernel | c -de |
| d bc-ad+bd+cd+d2+de |
2
o8 : R5-module, quotient of R5
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i9 : pdim P
o9 = 1
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i10 : Q = pushForward(F,P)
3
o10 = R4
o10 : R4-module, free, degrees {0..1, 0}
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i11 : pdim Q
o11 = 0
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Example: generic projection of a homogeneous coordinate ring
We compute the pushforward N of the homogeneous coordinate ring M of the twisted cubic curve in P^3.
i12 : P3 = QQ[a..d];
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i13 : M = comodule monomialCurveIdeal(P3,{1,2,3})
o13 = cokernel | c2-bd bc-ad b2-ac |
1
o13 : P3-module, quotient of P3
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The result is a module with the same codimension, degree and genus as the twisted cubic, but the support is a cubic in the plane, necessarily having one node.
i14 : P2 = QQ[a,b,c];
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i15 : F = map(P3,P2,random(P3^1, P3^{-1,-1,-1}))
5 2 5 6 5 9 3 3 3 5
o15 = map(P3,P2,{-a + 3b + -c + -d, -a + -b + -c + -d, -a + --b + 5c + -d})
8 7 3 7 4 5 5 8 10 4
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 118132779493238580ab-104320244778141000b2-280298569179375600ac+325552374370475700bc-181965012694790000c2 23626555898647716a2-6966317247015000b2-89961597776505420ac+81494680013145000bc-65571426590112400c2 1409568096125732028951004969567777734375b3-6482719265705705600371294601318907150000b2c-78087102433812475632538179594091656000ac2+7888421184326436734888423025409930800000bc2-864257762508175577163884613332068040000c3 0 |
{1} | -422839227297510408a+3367049722032303475b-617571100065483300c -219081460735415145a+1490548809846902750b-290848501074107420c 10403366126117550758849630601783252206397a2-100605531292696802494328224579263967197150ab+110208926487240747485741785280984734040625b2+32658302597352486447327293431767135547680ac-87289188756530224840415464276517565134900bc+17004516509742876927352956222565933446000c2 231969133211508a3-2448098997019200a2b+4550845921807500ab2-3065879120390625b3+881415693848340a2c-4321527799146600abc+5500587023032500b2c+884799243484800ac2-2201289868194000bc2+288601673680000c3 |
2
o16 : P2-module, quotient of P2
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i17 : hilbertPolynomial M
o17 = - 2*P + 3*P
0 1
o17 : ProjectiveHilbertPolynomial
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i18 : hilbertPolynomial N
o18 = - 2*P + 3*P
0 1
o18 : ProjectiveHilbertPolynomial
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i19 : ann N
3 2 2
o19 = ideal(231969133211508a - 2448098997019200a b + 4550845921807500a*b -
-----------------------------------------------------------------------
3 2
3065879120390625b + 881415693848340a c - 4321527799146600a*b*c +
-----------------------------------------------------------------------
2 2 2
5500587023032500b c + 884799243484800a*c - 2201289868194000b*c +
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3
288601673680000c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.